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A noncommutative version of Lie algebras: Leibniz algebras. (Une version non commutative des algèbres de Lie: les algèbres de Leibniz.) (French) Zbl 0806.55009
A Leibniz algebra over a commutative ring $$\mathbb{K}$$ is a $$\mathbb{K}$$-module $$g$$ together with a bilinear map $$[-,-]: g\times g\to g$$ satisfying for all $$x$$, $$y$$, $$z$$ in $$G$$, $$[x, [y,z]] - [[x, y],z]+ [[x, z],y]=0$$. The paper introduces this new algebraic structure and presents some interesting properties and connections with other theories. The main motivation is the existence of a homology theory $$HL_*$$ that gives in particular new algebraic invariants for Lie algebras.
This paper is clearly written and gives a lot of new and interesting questions in homological algebra.

##### MSC:
 17A32 Leibniz algebras 17B55 Homological methods in Lie (super)algebras 18G60 Other (co)homology theories (MSC2010) 19D55 $$K$$-theory and homology; cyclic homology and cohomology